Subtracting Polynomials: A Step-by-Step Guide

Welcome to our guide on subtracting polynomials! If you've ever stared at an equation like $\left(-6 s^2+12 s-8\right)-\left(3 s^2+8 s-6\right)=$ and felt a bit lost, you're in the right place. We're going to break down this process into simple, manageable steps, making polynomial subtraction feel like a breeze. Polynomials are fundamental building blocks in algebra, and mastering their subtraction is a key skill that will serve you well in more advanced mathematical concepts. Think of polynomials as algebraic expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication. When we subtract polynomials, we're essentially removing one expression from another. The core idea is to distribute the negative sign to each term in the second polynomial and then combine like terms. Don't worry if this sounds a bit abstract right now; we'll walk through it with the example provided and discuss the underlying principles. Our goal is to demystify this process, so whether you're a student tackling homework or just looking to refresh your math skills, you'll come away feeling confident. We'll cover everything from understanding what polynomials are, to the crucial step of distributing the negative sign, and finally, how to combine those pesky like terms to arrive at your simplified answer. By the end of this article, you'll be able to confidently solve problems like the one presented and understand why you're doing each step. So, grab a pen and paper, and let's dive into the world of polynomial subtraction!

Understanding the Basics: What are Polynomials?

Before we get our hands dirty with subtraction, let's make sure we're on the same page about what polynomials are. In essence, a polynomial is an algebraic expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, $3x^2 + 2x - 5$ is a polynomial. Here, $x$ is the variable, 3, 2, and -5 are the coefficients, and the exponents (2, 1, and 0 for the constant term) are non-negative integers. The terms are the parts of the polynomial separated by addition or subtraction signs. In $3x^2 + 2x - 5$, the terms are $3x^2$, $2x$, and $-5$. The degree of a term is the exponent of the variable, so $3x^2$ has a degree of 2, $2x$ has a degree of 1, and $-5$ (which can be written as $-5x^0$) has a degree of 0. The degree of the polynomial is the highest degree of any of its terms. In our example, the degree is 2.

When we talk about polynomials in the context of subtraction, we often encounter them in standard form, where the terms are arranged in descending order of their degrees. For instance, $12s - 8 - 6s^2$ is a polynomial, but its standard form is $-6s^2 + 12s - 8$. Similarly, $8s - 6 + 3s^2$ in standard form is $3s^2 + 8s - 6$. Understanding this standard form is helpful because it makes it easier to identify and combine 'like terms,' which is a critical step in polynomial operations. Like terms are terms that have the same variable(s) raised to the same power(s). For example, in the expression $5x^2 + 3x - 2x^2 + 7$, the terms $5x^2$ and $-2x^2$ are like terms because they both have the variable $x$ raised to the power of 2. The terms $3x$ and $7$ are not like terms with each other, nor are they like terms with $5x^2$ or $-2x^2$.

Now, let's consider the specific polynomials in our problem: $\left(-6 s^2+12 s-8\right)$ and $\left(3 s^2+8 s-6\right)$. The first polynomial, $-6s^2 + 12s - 8$, has three terms: $-6s^2$, $12s$, and $-8$. The second polynomial, $3s^2 + 8s - 6$, also has three terms: $3s^2$, $8s$, and $-6$. Notice that both polynomials are already in standard form, with terms ordered from highest degree to lowest degree. This makes our job of combining like terms much simpler later on. We will be subtracting the second polynomial from the first, and this involves carefully handling the signs of each term in the second polynomial. This foundational understanding of polynomials and their components is what allows us to proceed confidently with the subtraction operation.

The Crucial Step: Distributing the Negative Sign

The core of subtracting polynomials lies in correctly distributing the negative sign. When you see a minus sign in front of a polynomial enclosed in parentheses, like the $-\left(3 s^2+8 s-6\right)$ in our problem, it means you need to multiply each term inside those parentheses by -1. This is equivalent to changing the sign of every term within the second polynomial. This step is absolutely vital, and a common pitfall for many students is forgetting to change the sign of all the terms, or only changing the sign of the first term. Let's take our example: $\left(-6 s^2+12 s-8\right)-\left(3 s^2+8 s-6\right)$. The second polynomial is $3s^2 + 8s - 6$. When we apply the negative sign, each term inside the parentheses flips its sign:

• The term $3s^2$ becomes $-3s^2$.
• The term $8s$ becomes $-8s$.
• The term $-6$ becomes $+6$.

So, the expression $-\left(3 s^2+8 s-6\right)$ is equivalent to $-3s^2 - 8s + 6$. This transformation is key because it converts the subtraction problem into an addition problem, where we are adding the terms of the first polynomial to the terms of the negated second polynomial.

It's helpful to visualize this. Imagine you have a bag of items represented by the first polynomial, and you're being asked to take away another set of items represented by the second polynomial. Taking away items means you are removing them, and if those items had certain values (positive or negative), removing them changes the overall value. Distributing the negative sign mathematically achieves this 'removal' or 'inversion' of the values. If you had $+3s^2$ items and you take them away, you lose 3 positive $s^2$ items, hence you are left with $-3s^2$. If you had $-6$ items (a debt of 6) and you take them away, you are essentially removing a debt, which is like gaining 6 positive items, hence you are left with $+6$.

This distribution step is where many errors can occur. Always double-check that you have changed the sign of every single term within the parentheses that is being subtracted. A good mnemonic is to think of it as multiplying each term by $-1$. So, $(-1) \times (3s^2) = -3s^2$, $(-1) \times (8s) = -8s$, and $(-1) \times (-6) = +6$. Once this distribution is done correctly, the rest of the problem becomes much more straightforward. You'll rewrite the original expression with the distributed terms, setting the stage for the final step: combining like terms.

Combining Like Terms for the Final Answer

After successfully distributing the negative sign, our polynomial subtraction problem has transformed into an addition problem. We now have the expression $\left(-6 s^2+12 s-8\right) + \left(-3 s^2-8 s+6\right)$. The next and final step is to combine the like terms. Remember, like terms are terms that have the exact same variable part (same variable raised to the same power). In our example, the like terms are the $s^2$ terms, the $s$ terms, and the constant terms.

Let's group the like terms together from our expression: $-6s^2 + 12s - 8 - 3s^2 - 8s + 6$.

1. Combine the $s^2$ terms: We have $-6s^2$ and $-3s^2$. When we combine them, we add their coefficients: $-6 + (-3) = -6 - 3 = -9$. So, the combined $s^2$ term is $-9s^2$.

2. Combine the $s$ terms: We have $+12s$ and $-8s$. Adding their coefficients gives us: $12 + (-8) = 12 - 8 = 4$. So, the combined $s$ term is $+4s$.

3. Combine the constant terms: We have $-8$ and $+6$. Adding them together: $-8 + 6 = -2$. So, the combined constant term is $-2$.

Now, we put these combined terms back together to form our final simplified polynomial. The result is $-9s^2 + 4s - 2$. This is our final answer.

This process of combining like terms is essentially arithmetic with coefficients. You're not changing the variable part; you're just adding or subtracting the numerical values that precede them. It's like saying you have 12 apples and you take away 8 apples; you end up with 4 apples. Similarly, you have $-6$ $s^2$ items and you add $-3$ $s^2$ items, you end up with $-9$ $s^2$ items.

It's crucial to be meticulous during this stage. Ensure you're only combining terms that are actually alike. Don't try to combine an $s^2$ term with an $s$ term, or an $s$ term with a constant. Each type of term (based on its variable and exponent) is distinct and must be treated separately. If you started with polynomials that weren't in standard form, it's highly recommended to rearrange them into standard form before you begin distributing and combining. This visual organization significantly reduces the chances of errors and makes the entire process much smoother. The final polynomial should also be presented in standard form, with terms ordered from highest degree to lowest degree, just as we found: $-9s^2 + 4s - 2$.

This method applies to any polynomial subtraction problem. The key is the systematic approach: identify the polynomials, distribute the negative sign to the second polynomial, and then combine all like terms. By following these steps carefully, you can confidently tackle any polynomial subtraction challenge that comes your way.

Conclusion: Mastering Polynomial Subtraction

We've successfully navigated the process of subtracting polynomials, starting from understanding the components of a polynomial to the critical steps of distributing the negative sign and combining like terms. The problem $\left(-6 s^2+12 s-8\right)-\left(3 s^2+8 s-6\right)$ was solved by first rewriting it as $\left(-6 s^2+12 s-8\right) + \left(-3 s^2-8 s+6\right)$ after distributing the negative sign. Then, by grouping and combining like terms (the $s^2$ terms, the $s$ terms, and the constant terms), we arrived at the simplified expression $-9s^2 + 4s - 2$. This result corresponds to option B among the choices provided.

Remember, the key takeaways for mastering polynomial subtraction are precision and organization. Always ensure you distribute the negative sign to every term in the polynomial being subtracted. Take your time when combining like terms, paying close attention to the signs of the coefficients. If you find yourself making frequent errors, it might be helpful to write out each step explicitly, perhaps even color-coding like terms, until the process becomes second nature. Practice is, as always, your greatest ally in solidifying these algebraic skills.

Polynomials are a fundamental concept in algebra and appear in many areas of mathematics and science, from graphing functions to solving complex equations. Developing a strong understanding of operations like addition and subtraction with polynomials will build a solid foundation for tackling more advanced topics.

For further exploration and practice on polynomial operations, you might find resources on Khan Academy to be incredibly helpful. They offer detailed explanations, video tutorials, and practice exercises that can reinforce your understanding.